American Mathematician Wins Abel Prize

American mathematician John Milnor was awarded the Abel Prize in mathematics Wednesday. Among his many achievements, Milnor proved the existence of exotic seven-dimensional spheres. Robert Siegel talks with Julie Rehmeyer about his work. Rehmeyer writes for Science News and Wired Magazine.

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ROBERT SIEGEL, host:

A few months ago, when we took note of last year's winners of the Fields Medal in mathematics, Julie Rehmeyer, who writes about math and science, said that the Fields Medals are often called the Nobel Prize of mathematics. And her caution stems from the fact that while there is no Nobel for math, there is a prize, she told me, that comes a lot closer to being just that. And unlike the Fields Medal, this one is not limited to mathematicians under 40. It does carry an award of nearly a million dollars and it's presented by the king of Norway.

It is the Abel Prize and this year's winner is John Milnor, who's an American mathematician, now a distinguished professor at Stony Brook University in New York.

And Julie Rehmeyer joins us once again. Hi, welcome back.

Ms. JULIE REHMEYER (Math Columnist, ScienceNews.org): Hi, pleasure to be here.

SIEGEL: And I gather, John Willard Milnor is known for his work on exotic seven-dimensional spheres. I say that as if I have the faintest notion of what I'm talking about. For those of us who only think about three dimensions, explain what roughly does that mean?

Ms. REHMEYER: Okay. Imagine for a moment that you're a flat bug on a sheet of paper...

(Soundbite of laughter)

SIEGEL: All right.

Ms. REHMEYER: All right. And you can just move around on that paper and you would have no idea that there's another dimension out there, that there are three dimensions, because all you can see is what's on this piece of paper.

SIEGEL: Mm-hmm.

Ms. REHMEYER: Well, it could be that we're just like that, that we live in three dimensions, and we just can't see that there's another whole dimension or many whole dimensions out there. And one of the great challenges of mathematics is to understand what those high dimensional spaces are like, and what kinds of high dimensional objects there can be.

SIEGEL: So why spheres? What's so important about spheres?

Ms. REHMEYER: Spheres are a really great starting place because they're the very simplest objects out there. So if we can understand what is going on with spheres, then we've got a chance of figuring out what's going on with more complicated objects.

SIEGEL: We can imagine a two-dimensional sphere, it would just be a circle -would be...

Ms. REHMEYER: Yes. A sphere of two dimensions is a circle, and a sphere in three dimensions is like the surface of a ball.

SIEGEL: And after that we have to start thinking like mathematicians?

Ms. REHMEYER: Yeah. It gets harder to visualize after that. Actually, you can sort of visualize a sphere in four dimensions. Inside it it would be just like the world we live in, but if you walked, say, north and just kept going and going and going, eventually you'd get back to the same spot.

SIEGEL: I'm thinking about that.

(Soundbite of laughter)

SIEGEL: But we may not have enough time for me to digest the thought. So Professor Milnor, as I read on one website, proved the existence of seven-dimensional exotic spheres.

Ms. REHMEYER: That's right. So the amazing thing that he found was that in seven dimensions there are 28 different kinds of spheres. In all dimensions below that, there's only one kind of sphere, and nobody had any idea that there could be multiple kinds of spheres.

SIEGEL: And that is the main contribution of Professor Milnor?

Ms. REHMEYER: That's one of his most central, although he's had many other important contributions as well. He's a very wide-ranging mathematician. Another one of his contributions that's pretty interesting is called the hairy ball theorem.

SIEGEL: Hairy ball?

Ms. REHMEYER: Hairy ball theorem, yes. Imagine a beach ball covered with hair.

SIEGEL: Mm-hmm.

Ms. REHMEYER: And the question is, can you comb the hair on the beach ball so that it all lies flat, so that none of the hair is sticking up?

SIEGEL: Nothing is sticking up. Mm-hmm.

Ms. REHMEYER: Yeah. And the answer is that you can't do it. And what Milnor showed is that on a circle, or on a sphere in four-dimensional space, or a sphere in eight-dimensional space, you can do it, you can comb it flat. In every other dimension you can't. So it's pretty neat because, essentially, it's part of discovering the character of different dimensional spaces.

SIEGEL: And once again, the reason that we're hearing about John Milnor is that he is this year's winner of the Abel Prize for mathematics.

Julie Rehmeyer writes about mathematics in Berkeley, California. Thanks for talking with us once again.

Ms. REHMEYER: Thank you so much.

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